From 85daec38d024c8a5db293711622e892a24813a79 Mon Sep 17 00:00:00 2001 From: salagne Date: Fri, 17 Jul 2020 18:57:38 +0200 Subject: [PATCH] minor corrections and new bsp --- ana20.tex | 76 ++++++++++++++++++++++++++++++++++--------------- analysisII.pdf | Bin 1056671 -> 947614 bytes 2 files changed, 53 insertions(+), 23 deletions(-) diff --git a/ana20.tex b/ana20.tex index c8c3e53..459afac 100644 --- a/ana20.tex +++ b/ana20.tex @@ -4,7 +4,7 @@ \chapter{Kurven im \texorpdfstring{$\R^{n}$}{R\unichar{"207F}}} \section{Kurven} \begin{definition}[Kurve] - Eine Kurve $\gamma$ im $\R^n$ ist eine stetige Abbildung $\gamma\colon I\to \R^n$, $I$ Intervall (z.B. $I = [a,b]$ oder $I = \R$.) Schreibweise: \[\gamma(t) = \begin{pmatrix} + Eine Kurve $\gamma$ im $\R^n$ ist eine stetige Abbildung $\gamma\colon I\to \R^n$, $I$ Intervall (z.B. $I = [a,b]$ oder $I = \R$). Schreibweise: \[\gamma(t) = \begin{pmatrix} \gamma_1(t)\\ \vdots\\ \gamma_n(t) @@ -13,14 +13,15 @@ \end{definition} \begin{bsp} \begin{figure}[h] + \centering \captionsetup[subfigure]{justification=justified,singlelinecheck=false} \begin{subfigure}[b]{0.3\textwidth} \begin{tikzpicture} \draw[color=white] (-.5,-.5) -- (0,0); - \draw[->,color=blue, thick] (1,1) -- (2,2); + \draw[->,color=blue, thick] (0.5,0.5) -- node[above left] {$v$} (1.5,1.5); \draw (0,0) -- (2,2); - \node at (1,1) {\textbullet}; - \node[below] at (1,1) {$a$}; + \node at (0.5,0.5) {\textbullet}; + \node[below right] at (0.5,0.5) {$a$}; \end{tikzpicture} \subcaption{Beispiel 1, Gerade} \end{subfigure} @@ -83,6 +84,7 @@ \end{enumerate} \end{definition} \begin{figure} + \centering \begin{subfigure}[b]{0.4\textwidth} \begin{tikzpicture}[scale=0.7] \begin{axis}[axis lines=middle] @@ -164,7 +166,7 @@ \begin{definition}[Tangente] Sei $\gamma\in C^1(I;\R^n)$. Sei ein $t_0\in I$ regulär (d.h. $\gamma'(t_0) \neq 0$). Dann ist die Tangente an $\gamma(t_0)$ eine Gerade durch $\gamma(t_0)$ in Richtung $\gamma'(t_0)$ \[ - \gamma(t_0) + s\gamma'(t_0) \mid s\in \R\}. + \{\gamma(t_0) + s\gamma'(t_0) \mid s\in \R\}. \] \end{definition} \section{Die Bogenlänge} @@ -179,7 +181,7 @@ \] Seien $\mathcal{Z}_1$ und $\mathcal{Z}_2$ zwei Zerlegungen und $\mathcal{Z}^*$ eine gemeinsame Verfeinerung. Dann gilt $S(\mathcal{Z}^*) \geq \max(S(\mathcal{Z}_1), S(\mathcal{Z}_2))$ \begin{definition}[Rektifizierbarkeit] - Eine stetige Kurve $\gamma\in C^0(I;\R^n)$ heißt rektifizierbar, wenn die Menge aller Längen $S(\mathcal{Z})$ von Polygonen zu Partitionen $\mathcal{Z}$ von $I$ beschränkt ist. In diesem Fall heißt $S(\gamma) \colon \sup\{S(\mathcal{Z})\mid \mathcal{Z} \text{ Partition von } I\}$ die Länge von $\gamma$, in anderen Worten $\forall\epsilon > 0,\; \exists \delta > 0, \;\forall$ Partitionen von $I$ gilt: + Eine stetige Kurve $\gamma\in C^0(I;\R^n)$ heißt rektifizierbar, wenn die Menge aller Längen $S(\mathcal{Z})$ von Polygonen zu Partitionen $\mathcal{Z}$ von $I$ beschränkt ist. In diesem Fall heißt $S(\gamma) \coloneqq \sup\{S(\mathcal{Z})\mid \mathcal{Z} \text{ Partition von } I\}$ die Länge von $\gamma$, in anderen Worten $\forall\epsilon > 0,\; \exists \delta > 0, \;\forall$ Partitionen von $I$ gilt: \[ \max_i |t_{i-1} - t_i| < \delta \implies |S(\mathcal{Z}) - S(\gamma)| < \epsilon \] @@ -197,7 +199,7 @@ \left.\gamma\right|_{[s_{i-1}, s_i]} \in C^1([s_{i-1},s_i], \R^n),\quad \forall i = 1, \dots, M. \] Dann ist $\gamma$ rektifizierbar und hat die Länge \[S(\gamma) = \int_a^b \norm{\gamma'(t)}_2\d t = \sum_{i = 1}^{M}\int_{s_{i-1}}^{s_i} \norm{\gamma'(t)}_2 \d t.\] - Insbesondere hat der Graph einer $C_1$-Funktion $f\in C^1([a,b], \R), \quad \gamma_f(t)\colon (t, f(t))^T$ die Länge \[S(\gamma_f) = \int_a^b \sqrt{1 + f'(t)^2} \d t.\] + Insbesondere hat der Graph einer $C_1$-Funktion $f\in C^1([a,b], \R), \ \gamma_f(t)\coloneqq (t, f(t))^T$ die Länge \[S(\gamma_f) = \int_a^b \sqrt{1 + f'(t)^2} \d t.\] \end{satz} \begin{proof} Sei $\mathcal{Z} = \{t_0, \dots, t_N\}$ eine Partition, betrachte $\mathcal{Z}^* = \mathcal{Z} \cup \{s_0,\dots, s_M\} = \{x_0,\dots, x_K\}$. Dann gilt @@ -213,7 +215,7 @@ \[ S(\mathcal{Z}) \geq \int_a^b \norm{\gamma'(t)} \d t - \epsilon. \] - Fixiere dazu ein $\epsilon > 0$. Wähle dann eine Treppenfunktion $\varphi$ auf $[a,b]$ mit \[\norm{\gamma'(t) - \varphi(t)} \le \frac{\epsilon}{2(b-a)} \forall t\in [a,b]\setminus\{s_0,\dots, s_M\}\] + Fixiere dazu ein $\epsilon > 0$. Wähle dann eine Treppenfunktion $\varphi$ auf $[a,b]$ mit \[\norm{\gamma'(t) - \varphi(t)} \le \frac{\epsilon}{2(b-a)} \quad \forall t\in [a,b]\setminus\{s_0,\dots, s_M\}\] ($\varphi$ existiert, weil $\gamma'(t)$ stückweise stetig ist). Wähle ferner eine (feine) Partition $a = t_0< t_1< \dots < t_N = b$, s.d. $\varphi\big|_{[t_{i-1}, t_i]}$ konstant ist $\forall i = 1,\dots, N$. Dann gilt nämlich \begin{salign*} S(Z) &= \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \gamma'(t) \d t}\\ @@ -223,30 +225,28 @@ &\geq \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \varphi(t) \d t} - \sum_{i = 1}^{N}\frac{\epsilon}{2(b-a)}|t_i-t_{i-1}|\\ &= \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \varphi(t) \d t} - \frac{\epsilon}{2}\\ &\stackrel{\varphi(t) = \mathrm{const}}{=} \sum_{i = 1}^{N}\int_{t_{i-1}}^{t_i} \norm{\varphi(t)} \d t - \frac{\epsilon}{2}\\ - &= \sum_{i = 1}^{N}\norm{\gamma'(t) + (\varphi(t) - \gamma'(t))} \d t - \frac{\epsilon}{2}\\ + &= \sum_{i = 1}^{N}\int_{t_{i-1}}^{t_i}\norm{\gamma'(t) + (\varphi(t) - \gamma'(t))} \d t - \frac{\epsilon}{2}\\ &\stackrel{\text{analog}}{\geq} \sum_{i = 1}^{N}\int_{t_{i-1}}^{t_i} \norm{\gamma'(t)} \d t - \epsilon = \int_a^b\norm{\gamma'(t)} \d t - \epsilon \end{salign*} Also existiert für ein beliebiges $\epsilon > 0$ eine Zerlegung $\mathcal{Z}$ mit $S(\mathcal{Z}) \geq \int_a^b\norm{\gamma'(t)}\d t - \epsilon$. Zusammen mit $S(y) \leq \int_a^b\norm{\gamma'(t)}\d t$ folgt $S(\gamma) = \int_a^b \norm{\gamma'(t)} \d t$. \end{proof} \begin{figure}[h] + \centering \captionsetup[subfigure]{justification=justified,singlelinecheck=false} \begin{subfigure}[b]{0.3\textwidth} \begin{tikzpicture} \draw[color=white] (-.5,-.5) -- (0,0); - \draw[color=black] (1.5,0) arc [start angle=0, end angle=200, radius=1.5] -- node[pos=.5, below] {$r$} (0,0); - \draw[color=blue] (1.5,0) arc [start angle=0, end angle=200, radius=1.5]; - \draw[color=black] (.4,0) arc [start angle=0, end angle=200, radius=.4]; - \draw (0,0) -- node[pos=.5,below] {$r$} (1.5,0); - \node[color = blue] at (1.6,.4) {$\gamma$}; - \node at (0,.2) {$\varphi$}; + \draw[color=black] (1.5,0) -- node[below] {$r$} (0,0) -- node[below] {$r$} (200:1.5); + \draw[color=black] (.4,0) arc [start angle=0, end angle=200, radius=.4] node[pos=0.45, below] {$\varphi$}; + \draw[color=blue] (1.5,0) arc [start angle=0, end angle=200, radius=1.5] node[near start, right] {$\gamma$}; \end{tikzpicture} \subcaption{Beispiel 1: Kreisbogen} \end{subfigure} \begin{subfigure}[b]{0.6\textwidth} \begin{tikzpicture} - \begin{axis}[axis equal image, axis lines=middle,width=\textwidth, xticklabels={0, $\pi$, $2\pi$}, xtick={0,3.14,6.28}, ymin=0,ymax=2, smooth] + \begin{axis}[axis equal image, axis lines=middle, xticklabels={0, $\pi$, $2\pi$}, xtick={0,3.14,6.28}, ymin=0,ymax=2.2, xmax=6.5, smooth] \addplot[domain=0:6.28] ({x-sin(180/3.14 * x)},{1-cos(180/3.14 * x)}); - \draw (3.14,1) circle (1); + \draw (3.14,1) circle [radius=1]; \node at (3.14,2) {\textbullet}; \end{axis} \end{tikzpicture} @@ -269,13 +269,14 @@ \end{pmatrix}$ und daher $\norm{\gamma'(t)}_2^2 = 1 - 2 \cos(t) + \cos^2(t) + \sin^2(t) = 2 - 2\cos(t) = 4 \sin^2(\frac{t}{2})$. Insgesamt gilt also \[ - S(\gamma) = \int_0^{2\pi} \underbrace{\left|2\sin\left(\frac{t}{2}\right)\right|}_{\geq 0} \d t \oldstackrel{x = \frac{t}{2}}{=} 4\int_0^\pi \sin(x) \d x = 8\] + S(\gamma) = \int_0^{2\pi} \left|\smash[b]{\underbrace{2\sin\left(\frac{t}{2}\right)}_{\geq 0}}\right| \d t \oldstackrel{x = \frac{t}{2}}{=} 4\int_0^\pi \sin(x) \d x = 8 + \vphantom{\underbrace{2\sin\left(\frac{t}{2}\right)}_{\geq 0}}\] \end{enumerate} \end{bsp} \section{Parametertransformationen} \begin{definition}[Parametertransformation] \begin{enumerate} - \item Sei $\varphi\colon [\alpha, \beta]\! \to\! [a,b]$ eine $C^k$-Abbildung $(k\in \N_0 \cup \infty_+)$ zwischen den Intervallen $[\alpha,\beta]$ und $[a,b]$, + \item Sei $\varphi\colon [\alpha, \beta]\! \to\! [a,b]$ eine $C^k$-Abbildung $(k\in \N_0 \cup +\infty)$ zwischen den Intervallen $[\alpha,\beta]$ und $[a,b]$, sei außerdem $\varphi$ bijektiv und $\varphi^{-1} \in C^k([a,b],\;[\alpha, \beta])$. Dann heißt $\varphi$ eine $C^k$-Parametertransformation. \item Sei weiter $\gamma\colon [a,b]\to\R^n$ eine Kurve. Dann heißt die Kurve $\delta:[\alpha, \beta] \to \R^n,\;\delta \coloneqq \gamma\circ \varphi$ die Umparametrisierung von $\gamma$ (mittels $\varphi$). @@ -290,8 +291,8 @@ S(\gamma\circ \varphi) &= \int_\alpha^\beta \norm{(\gamma\circ \varphi)'(\tau)} \d \tau\\ &\stackrel{\text{Kettenregel}}{=} \int_\alpha^\beta\norm{\gamma'(\varphi(\tau)) \cdot \varphi'(\tau)} \d \tau\\ &= \begin{cases} - \int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) > 0, \tau \in [a,b]\\ - -\int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) < 0, \tau \in [a,b]\\ + \int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) > 0, \tau \in [\alpha, \beta]\\ + -\int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) < 0, \tau \in [\alpha, \beta]\\ \end{cases}\\ &\stackrel{\substack{t=\varphi(\tau)\\\d t = \varphi'(\tau)\d \tau}}{=} \begin{cases} @@ -301,9 +302,9 @@ &= \int_a^b\norm{\gamma'(t)} \d t\\ &= S(\gamma) \end{salign*} - \item Umparametrisierung auf Bogenlänge. Sei $\gamma\colon [a,b] \to \R^n$ eine reguläre $C^1$-Kurve, d.h. $\gamma'(t) \neq 0\forall t\in [a,b]$. Definiere die Abbildung $\sigma\colon [a,b] \to [0, S(\gamma)]$ durch + \item Umparametrisierung auf Bogenlänge. Sei $\gamma\colon [a,b] \to \R^n$ eine reguläre $C^1$-Kurve, d.h. $\gamma'(t) \neq 0 \ \forall t\in [a,b]$. Definiere die Abbildung $\sigma\colon [a,b] \to [0, S(\gamma)]$ durch \[ - \sigma(t) \coloneqq \int_a^t\norm{\gamma'(\tau)} \d \tau \left(= S(\gamma(t))\bigg|_{[a,t]}\right). + \sigma(t) \coloneqq \int_a^t\norm{\gamma'(\tau)} \d \tau \left(= S\left(\gamma(t)\big|_{[a,t]}\right)\right). \] Wir können zeigen, dass $\varphi\coloneqq \sigma^{-1}$ eine orientierungstreue $C^1$-Parametertransformation ist und für die (\glqq auf Bogenlänge\grqq) umparametrisierte Kurve $\beta\colon [0,S(\gamma)]\to \R^n$ gilt \[S\left(\beta\big|_{[0,x]}\right) = x,\; \norm{\beta'(x)} = 1,\; \forall x\in [0,S(\gamma)].\] \begin{proof} @@ -315,4 +316,33 @@ \end{proof} \end{enumerate} \end{bem} + + \begin{bsp} + Umparametrisierung auf Bogenlänge einer Zykloide: Es gilt $\gamma(t) = (t-\sin t, 1-\cos t)^T$ und damit + \begin{align*} + \norm{\gamma'(t)} &= 2\sin\left(\frac{t}{2}\right), \qquad 0\le t \le 2\pi\\ + \implies \norm{\gamma'(t)} &> 0 \qquad \text{für } 2\varepsilon \le t \le 2\pi-2\varepsilon, \ \varepsilon > 0 + \end{align*} + Betrachte also $\gamma \colon [2\varepsilon, 2\pi-2\varepsilon] \to \R^2$. Wir definieren + \begin{salign*} + \sigma(t) &= S\left(\gamma\big|_{[2\varepsilon,t]}\right)\\ + &= \int_{2\varepsilon}^t \norm{\gamma'(\tau)} \d \tau \\ + &= 2\int_{2\varepsilon}^t \sin\left(\frac{\tau}{2}\right) \d \tau \\ + &\stackrel{\substack{s=\tau/2\\\mathrm{d} s=\mathrm{d}\tau/2}}{=} 4\int_{\varepsilon}^{t/2} \sin s \d s\\ + &= -4\cos s\big|_{\varepsilon}^{t/2} + = 4\left(\cos\varepsilon - \cos\frac{t}{2}\right) + \end{salign*} + Somit gilt $\sigma \colon [2\varepsilon,2\pi-2\varepsilon] \to [0,8\cos\varepsilon]$. Ziel: + \begin{align*} + \varphi \colon [0,8\cos\varepsilon] &\to [2\varepsilon,2\pi-2\varepsilon]\\ + s &\mapsto \varphi(s)=t + \end{align*} + Dazu setzen wir $\varphi = \sigma^{-1}$ und bestimmen die Umkehrfunktion von $\sigma$ + \begin{align*} + s &= 4\left(\cos\varepsilon - \cos\frac{t}{2}\right)\\ + \implies \cos\frac{t}{2} &= \cos\varepsilon - \frac{s}{4}\\ + \implies t &= \underbrace{2 \arccos\left(\cos\varepsilon-\frac{s}{4}\right)}_{\varphi\mathrlap{(s), \ s \in [0,8\cos\varepsilon]}} + \end{align*} + Insgesamt erhalten wir $\beta(s) = \gamma(\varphi(s))$. + \end{bsp} \end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 3803b1d2f0c83cc2def8d1423a7b243354ac3b74..253fb98af78f5396e6c66fa69c8beb64b3509cdc 100644 GIT binary patch delta 697877 zcmV)YK&-!?zd)YrIglg+F*YEVfjQ0{&M#CneTk)vszGU}DN58?&f#1R_$%GeIcNfpHcbug%ae zW7MoWQxC!V&Fx$3HIyZIBA((R1C{fjI>GLKXzS2_u5sH0^g|+(rNe)8fE%Il4Tt?O 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